Linear Algebra with the help of SAGE

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Linear Algebra with the help of SAGE
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What is SAGE ?

• free open source mathematics software system.
• Visit www.sagemath.org and open a free account .
• Have a web version, so no need to download and
  install !!

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Some Basics

• xvar(x) tells the sage to recognize x as a
  symbolic variable.
• Typing Command?? displays the syntax and
  examples for the required command.
• Can save the worksheet for future reference.
• Keyboard shortcut for evaluate is shiftreturn.

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Matrix Notations

• Amatrix(RDF,1,2,4,5) defines a matrix of
  order 2 by 2 over the field of real numbers.
• Operations like matrix addition, multiplication
  by matrix or scalar are defined as usual.

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Determinant and Inverse

• Det(A) gives the determinant of a matrix, A.
• A.inverse() gives the inverse of A.

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LU-Decomposition

• P,L,UA.LU() give a tuple of 3 matrices
  corresponding to P, L, U resp.
• P is the permutation matrix.
• L is the lower triangular matrix.
• U is the upper triangular matrix.
• Here PALU.

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Rank of a matrix

• Rank(A) gives the rank of matrix A.
• A.nullity() gives the dimension of null space of
  A.
• For dimension of left kernel/null space use
  A.left_nullity().

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Eigen Values

• A.eigenvalues() gives the Eigen values.
• A.eigenvectors_left() returns left Eigen vectors.
• A.eigenspaces() gives dimension of Eigen space
  for each Eigen value.

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Some more ..

• B.echelon_form() gives the reduced Echelon form
  of B.
• B.row_space() gives the basis and dimension of
  row space of B.

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Help

• Type A. and the press Tab key. It lists all
  sorts of nice things we can do with matrices.